We begin in §2 by considering families of cubic fields with one fixed invariant. Invoking work of Bhargava–Taniguchi–FT [1]} on the Davenport-Heilbronn theorem, we obtain asymptotics for families of cubic fields with fixed \(\mathrm {F}\) ; using work of Cohen–Morra [2]} and Cohen–FT [3]} on a Kummer-theoretic approach, we deduce asymptotics for families of cubic fields with fixed \(\mathrm {D}\) . The leading constants appearing in the asymptotics for the latter family are somewhat inexplicit, but in §3 we prove that the average values of these constants have an explicit description given in terms of products of mass formulae.
